Reproductive strong solutions of Navier-Stokes equations with non homogeneous boundary conditions

Chérif Amrouche (1) , Macaire Batchi (1, 2) , Jean Batina (2)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 LATEP - Laboratoire de Génie Thermique Énergétique et Procédés (EA1932)

The object of the present paper is to show the existence and the uniqueness of a reproductive strong solution of the Navier-Stokes equations, i.e. the solution $\boldsymbol{u} $ belongs to $\text{}\mathbf{L}% ^{\infty }\left( 0,T;V\right)\, \cap \,\mathbf{L}^{2}\left( 0,T;\mathbf{H}% ^{2}\left( \Omega \right) \right)$ and satisfies the property $\boldsymbol{u}% \left( \boldsymbol{x,\,}T\right) =\boldsymbol{u}% \left( \boldsymbol{x,\,}0\right) =\boldsymbol{u}_{0}\left( \boldsymbol{x}\right)$. One considers the case of an incompressible fluid in two dimensions with nonhomogeneous boundary conditions, and external forces are neglected.

Mots clés: Navier-Stokes equations; incompressible fluid; reproductive solution; nonhomogeneous boundary conditions

Domaines: Equations aux dérivées partielles [math.AP]

Chérif Amrouche, Macaire Batchi, Jean Batina. Reproductive strong solutions of Navier-Stokes equations with non homogeneous boundary conditions. 2007. ⟨hal-00142028⟩